The correct answer is towards the north.

Key Points

• A proton moves towards east by entering an uniform magnetic field in the downward direction. so, the proton is deflected in a mutually perpendicular direction to the magnetic field.
• Using Fleming's left hand rule, it is towards the north direction.

Additional Information

• Fleming's left hand rule states stretching the forefinger, middle finger and thumb in such a way that they are mutually perpendicular to each other.
• Forefinger points in the direction of the magnetic field, the middle finger points in the direction of current and the thumb points in the direction of force.

Given:

44 ÷ 120 × 5 - 2 + 10 × 2 = ?

Concept:

According to the given question,

Replacing the given sign in the given problem, we get:

44 + 120 ÷ 5 × 2 - 10 ÷ 2 = ?

According to BODMAS rule,

44 + 120 ÷ 5 × 2 - 10 ÷ 2 = ?

= 44 + 24 × 2 - 5

= 44 + 48 - 5

= 92 - 5 

= 87

Shortcut Trick

Put θ = 45° to simplify the expression directly.

Numerator: sin(45°) × cos(45°) × cot(45°) = (1/√2) × (1/√2) × 1 = 1/2.

Denominator: cos²(45°) − 1 = (1/√2)² − 1 = 1/2 − 1 = −1/2.

Result: (1/2) ÷ (−1/2) = −1.

∴ The correct answer is -1.

Alternate Method

Given: Expresssion = [sin(90°−θ) × cos(90°−θ) × cot(90°−θ)] / [cos²θ − 1]

Formula Used: sin(90°−θ) = cosθ; cos(90°−θ) = sinθ; cot(90°−θ) = tanθ; sin²θ + cos²θ = 1

⇒ Numerator = cosθ × sinθ × tanθ

⇒ Numerator = cosθ × sinθ × (sinθ / cosθ) = sin²θ

⇒ Denominator = cos²θ − 1 = −(1 − cos²θ) = −sin²θ

⇒ Final Value = sin²θ / (−sin²θ) = −1

∴ The correct answer is -1.

Additional Information

Complementary Angle Identities

Functions change to their co-functions in the first quadrant: sin ↔ cos, tan ↔ cot, and sec ↔ cosec.

Pythagorean Identities

The fundamental identity is sin²θ + cos²θ = 1, which implies cos²θ − 1 = −sin²θ and sin²θ − 1 = −cos²θ.

Given:

The position of Mahima from the front end is 28th

The position of Anshul from the rear end is 22nd

People in between Mahima and Anshul = 5

According to the given information, the diagram will be,

Total number of people present in Queue: 28 + 5 + 22 = 55

The correct answer is: '55'.

Shortcut Trick

Identify the expression as a difference of squares: (8x³)² − (y³)².

Apply a² − b² = (a − b)(a + b) to get (8x³ − y³)(8x³ + y³).

Recognize 8x³ as (2x)³ and factorize both the difference and sum of cubes.

The factors are (2x − y)(4x² + 2xy + y²) and (2x + y)(4x² − 2xy + y²).

∴ The correct answer is (2x − y)(2x + y)(4x² + 2xy + y²)(4x² − 2xy + y²).

Alternate Method

Given: 64x⁶ − y⁶

Formula Used:

1. a² − b² = (a − b)(a + b)
2. a³ − b³ = (a − b)(a² + ab + b²)
3. a³ + b³ = (a + b)(a² − ab + b²)

Calculations:

⇒ 64x⁶ − y⁶ = (8x³)² − (y³)²

⇒ (8x³ − y³)(8x³ + y³)

⇒ [(2x)³ − y³] × [(2x)³ + y³]

⇒ [(2x − y)((2x)² + (2x)(y) + y²)] × [(2x + y)((2x)² − (2x)(y) + y²)]

⇒ (2x − y)(4x² + 2xy + y²) × (2x + y)(4x² − 2xy + y²)

⇒ (2x − y)(2x + y)(4x² + 2xy + y²)(4x² − 2xy + y²)

∴ The correct answer is (2x − y)(2x + y)(4x² + 2xy + y²)(4x² − 2xy + y²).

Additional Information

Difference of Squares

An algebraic expression of the form a² − b² can always be factorized into (a + b)(a − b).

Difference and Sum of Cubes

a³ − b³ = (a − b)(a² + ab + b²) and a³ + b³ = (a + b)(a² − ab + b²) are essential for higher power factorizations.

Power of a Power Rule

x^mn can be written as (x^m)ⁿ or (xⁿ)^m, which helps in identifying square or cube forms.

The correct answer is scattering of light.

Key Points

• When white light from the sun enters the earth's atmosphere, the light gets scattered i.e. the light spreads in all directions by dust particle, free water molecule, and molecules of gases present in the atmosphere. This is called the scattering of light.
• During the scattering of light, some of the light is absorbed and the rest is scattered.

Additional Information

• When white light passes through the glass prism it splits into the seven constituent colors, this phenomenon is called dispersion of light.
• When a ray of light reaches a smooth polished surface it bounces back, it is called reflection of light.
• The bending of light as it passes from one substance to another, it is called refraction of light.

The correct answer is Composition.

Key Points

• The composition of trade refers to the types of goods and services a country exports and imports.
• It encompasses the variety and complexity of products and services exchanged in international trade.
• The composition can be determined by various factors, including a country's resources, technological capabilities, industrial structure, and policies.
• It reflects a country's economic strength and competitiveness on the global stage.
• Countries with a diverse trade composition are usually more resilient to economic shocks because they don't rely on a single type or category of goods for their export earnings.
• Over time, the composition of trade can change as a country develops, diversifies its economy, or adopts new technologies.

Additional Information

Shortcut Trick

Apply the successive percentage change formula: Net Change = a + b + (a × b)/100

Here, price decrease (a) = −35% and sales increase (b) = +20%.

Net Change = −35 + 20 + (−35 × 20)/100

Net Change = −15 − 700/100 = −15 − 7 = −22%

Negative sign indicates a decrease.

∴ The correct answer is 22% decrease.

Alternate Method

Given:

Price change = −35% | Sales change = +20%

Formula: Revenue = Price × Quantity Sold

Let Initial Price = 100 and Initial Sales = 100

⇒ Initial Revenue = 100 × 100 = 10000

⇒ New Price = 100 − 35% of 100 = 65

⇒ New Sales = 100 + 20% of 100 = 120

⇒ New Revenue = 65 × 120 = 7800

⇒ Change in Revenue = 7800 − 10000 = −2200

⇒ Percentage Change = (−2200 / 10000) × 100 = −22%

∴ The correct answer is 22% decrease.

Additional Information

Product Stability Rule

When the product of two variables (Price × Consumption) is constant, if one increases by r%, the other must decrease by [r/(100+r)] × 100%.

Successive Percentage Formula

For two successive changes of x% and y%, the net effective change is given by [x + y + (xy/100)]%.

Revenue and Price Elasticity

Revenue increases only if the percentage increase in quantity sold is greater than the percentage decrease in price.

Shortcut Trick

To find the nature of roots, calculate the discriminant (D) = b² − 4ac.

For the equation 5x² + 7x + 4 = 0, we have a = 5, b = 7, and c = 4.

D = 7² − 4 × 5 × 4 = 49 − 80 = −31.

As D < 0, the roots are imaginary and not real.

∴ The correct answer is No real roots.

Alternate Method

Given: The quadratic equation is 5x² + 7x + 4 = 0.

Formula Used:

The standard form of a quadratic equation is ax² + bx + c = 0.

The nature of roots is determined by the Discriminant (D):

1. If D > 0, roots are real and unequal.
2. If D = 0, roots are real and equal.
3. If D < 0, roots are not real (imaginary).

Where D = b² − 4ac.

Calculations:

Comparing 5x² + 7x + 4 = 0 with ax² + bx + c = 0:

⇒ a = 5

⇒ b = 7

⇒ c = 4

⇒ D = b² − 4ac

⇒ D = (7)² − 4 × 5 × 4

⇒ D = 49 − 80

⇒ D = −31

Since the value of the discriminant is less than zero (−31 < 0), the equation has no real roots.

∴ The correct answer is No real roots.

Additional Information

Discriminant and Perfect Squares

If a, b, and c are rational and D is a perfect square, the roots are rational; otherwise, they are irrational.

Nature of Roots Summary

D > 0 → Real and distinct | D = 0 → Real and equal | D < 0 → Imaginary roots.

Quadratic Formula

The roots of ax² + bx + c = 0 are calculated as x = (−b ± √D) / 2a.

Sum and Product of Roots

Sum of roots (α + β) = −b/a and Product of roots (αβ) = c/a.

The given series can be represented as:

So, there are 2 symbols: & and # in the series which are between the seventh element from the right end and the fifth element from the left end of the given series.

Hence, the correct answer is '2'.

The correct answer is Brahmaputra river.

Key Points

• Majuli in the Brahmaputra river is the largest inhabited riverine island in the world.
• In 2016, it became the first island to be a district in India.
• The island is formed by the Brahmaputra River in the south and the Kherkutia Xuti, an anabranch of the Brahmaputra River, joined by the Subansiri River in the north.
• It was created by changes in the flow of the Brahmaputra River and its tributaries, mostly the Lohit River.

Shortcut Trick

Directly expand the quadratic equation using the property (x + a)(x + b) = x² + (a + b)x + ab.

Here, a = −4 and b = 1, so the LHS becomes: x² + (−4 + 1)x + (−4 × 1) = x² − 3x − 4.

Equating to the RHS: x² − 3x − 4 = 2.

Shift the constant from the RHS to the LHS: x² − 3x − 4 − 2 = 0.

∴ The correct answer is x² − 3x − 6 = 0.

Alternate Method

Given: (x − 4)(x + 1) = 2

Method: Use the distributive property to expand the binomials and write in the standard form ax² + bx + c = 0.

⇒ x(x + 1) − 4(x + 1) = 2

⇒ x² + x − 4x − 4 = 2

⇒ x² − 3x − 4 = 2

⇒ x² − 3x − 4 − 2 = 0

⇒ x² − 3x − 6 = 0

∴ The correct answer is x² − 3x − 6 = 0.

Additional Information

Standard Form of Quadratic Equation

The standard representation of any quadratic equation is given by ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Nature of Roots

The nature of the roots can be determined by the discriminant (D), calculated as D = b² − 4ac.

Quadratic Formula

To find the roots of the equation ax² + bx + c = 0, use the formula x = [−b ± √(b² − 4ac)] / 2a.

The logic followed here is:

Hence, ‘23’ will replace the question mark (?) in the given series.

The correct answer is 'The virus can be transmitted by shaking hands with a person infected with HIV.'

Key Points

• AIDS virus (HIV Virus) is spread through body fluids such as blood, semen, rectum fluids, vaginal fluids and breast milk.
• The virus can be transmitted when these fluids in an infected person come into contact with mucous membranes in the rectum, vagina, penis, or mouth of another person.
• It cannot spread through touch, food, saliva, mosquitos and water.
• Thus, a person cannot get infected by shaking hands with a person infected with HIV.

Additional Information

• ​ Various modes of transmission of HIV virus are: 
• By sharing the syringes used for injecting drugs.
• Through sexual contact with an infected person.
• Through breast milk to an infant.

Given:

Anu, Dilip, Eshwar, Jahanvi, Komal, Pallavi, Riya, and Tara are sitting around a square table facing the centre of the table.

Some of them are sitting at the corners, while some are sitting at the exact centre of the sides.

According to the given information,

(1.) Tara is to the immediate right of Komal and to the immediate left of Jahanvi.

 (2.) Eshwar is sitting at the exact centre of one of the sides and has Riya to the immediate left and Komal to the immediate right.

• Eshwar has Riya to the immediate left and Komal to the immediate right.

(3.) Anu is to the immediate left of Riya.

(4.) Dilip and Jahanvi have only one person between them.

Hence, 'Dilip' is sitting to the immediate right of Pallavi.

The correct answer is '15'.

Key Points

• Non-metallic elements are the elements with high ionization energies, high electronegativity and small size.
• They gain electrons on reaction with other compounds forming covalent bonds.
• Owing to all these properties, non-metals are placed in Group- 14,15,16 and 17.
• But in group 14 there is only one non-metal present which is Carbon.

Additional Information

• No. of non-metals in group 15 - Two
• No. of non-metals in group 16 - Three
• No. of non-metals in group 17 - Four

​

The correct answer is 'Iris'.

Key Points

• Iris is responsible for giving colors to the eyes.
• If there is less melanin in the iris, light can pass through and is reflected by collagen fibres that are located deeper in the eye. The colour of the eyes becomes bluer when light is reflected back.
• The color can range from light blue to dark brown, depending on the degree of pigmentation.

Additional Information

• Iris is also responsible for the constriction and enlargement of pupil which decides how much light will pass through it.
• Functions of other parts of eyes:
• Lens: Transparent biconvex surface which refracts light focusing it onto the retina.
• Retina: It receives the light which lens has focused, convert the light into neural signals and send these signals to the brain.
• Cornea: It controls and focuses the entry of light into the eyes.

According to the given information, we have the possible arrangements as follows:

1). Anant is taller than Gagan $\longrightarrow$ Anant > Gagan

2). Manoj is shorter than Janak $\longrightarrow$ Janak > Manoj 

3). Parth is taller than Janak but shorter than Gagan $\longrightarrow$ Parth > Janak < Gagan

4). If Anurag is the shortest, then the Final arrangement should be like $\longrightarrow$  Anant > Gagan > Parth > Janak > Manoj > Anurag

Hence, "Anant" is the tallest among them.

Shortcut Trick

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Ratio of sides = 5 : 6

Ratio of areas = 5² : 6² = 25 : 36

∴ The correct answer is 25 : 36.

Alternate Method

Given:

Ratio of corresponding sides (s₁ : s₂) = 5 : 6

Formula Used:

Area(Δ1) / Area(Δ2) = (s₁ / s₂)²

Calculations:

⇒ Let the sides be 5k and 6k.

⇒ Area Ratio = (5k / 6k)²

⇒ Area Ratio = 25k² / 36k²

⇒ Area Ratio = 25 : 36

∴ The correct answer is 25 : 36.

Additional Information

Area of Similar Triangles Theorem

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding altitudes, medians, or angle bisectors.

Perimeter Ratio

The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides (P₁/P₂ = s₁/s₂).

Similarity Criteria

Triangles are similar if they satisfy AA (Angle-Angle), SAS (Side-Angle-Side), or SSS (Side-Side-Side) conditions.

Shortcut Trick

Since the Curved Surface Area (CSA) remains constant, the product of radius (r) and height (h) must be constant (r₁h₁ = r₂h₂).

Given: r₁ = 4 cm, h₁ = 4 cm, r₂ = 16 cm.

4 × 4 = 16 × h₂

h₂ = 16 ÷ 16 = 1 cm.

∴ The correct answer is 1 cm.

Alternate Method

Given: Initial radius (r₁) = 4 cm, Final radius (r₂) = 16 cm, Initial height (h₁) = 4 cm.

Formula Used: Curved Surface Area of Cylinder = 2πrh

⇒ According to question: CSA_initial = CSA_final

⇒ 2 × π × r₁ × h₁ = 2 × π × r₂ × h₂

⇒ r₁ × h₁ = r₂ × h₂

⇒ 4 × 4 = 16 × h₂

⇒ 16 = 16 × h₂

⇒ h₂ = 1 cm

∴ The correct answer is 1 cm.

Additional Information

Total Surface Area of Cylinder

The total area including bases is given by TSA = 2πr(h + r).

Volume of Cylinder

The space occupied by a cylinder is V = πr²h.

Inverse Relationship

For a constant area, if the radius is multiplied by n, the height must be divided by n.

The correct answer is Cuboidal epithelium.

Key Points

• Cuboidal epithelium forms the lining of kidney tubules and duets of salivary glands.
• It provides mechanical support to the tubules and the glands.
• Cuboidal epithelium is cubical or more or less squarish in shape.
• The Nucleus of the cells lies in the centre.

Additional Information

• Locations of all other epithelium are as follows: 
• Squamous epithelium: Is present in walls of capillaries, air sacs, alveoli, lungs, and the kidney.
• Columnar epithelium: Lines the surface of the cervix and endocervical crypts which helps in absorption and secretion.
• Glandular epithelium: Forms covering all the major glands.

The correct answer is 7^th century.

Key Points

• Islam is, major world religion promulgated by the Prophet Muhammad in Arabia in the 7th century CE.
• Islam Muhammad is considered the last of a series of prophets.
• Islam teaches that the creation of everything in the universe was brought into being by God's command.
• Adherents of Islam, called Muslims, number approximately 1.8 billion globally and are the world's second-largest religious population behind Christians.

The correct answer is Squash.

Key Points

• Dipika Pallikal won the women's doubles and mixed doubles titles with Joshna Chinappa and Saurav Ghosal, respectively at the World Doubles Squash Championships in Glasgow on 9 April 2022.
• Dipika Pallikal and Saurav Ghosal defeated England's Adrian Waller and Alison Waters 11-6, 11-8 in the final at the Scotstoun Leisure Centre.
• They became India's first-ever World Doubles squash champions.

Additional Information 

• ​Anahat Singh has become the first Indian girl to win the U-15 girls category of the Junior US Open Squash tournament in Philadelphia.
• Saurav Ghosal has become the first Indian squash player to win the Malaysian Open Championships in November 2021.

Statements:

All anklets are printers.

Some printers are birds.

Some birds are cherries.

The least possible Venn diagram for the given statements is as follows:

Conclusions:

I. Some birds are printers ⇒ True (As Some printers are birds so, from the above Venn diagram we can conclude that some birds are printers.)

II. All printers are anklets.⇒ False (As all anklets are printers but vice-versa is not true in a definite sense.)

III. Some cherries are birds ⇒True (As Some birds are cherries so, from the above Venn diagram we can conclude that Some cherries are birds.)

Hence, "Both conclusions I and III follow".

Shortcut Trick

Identify the multiplication factor between the first term and the third term.

Since 10 × 3 = 30, apply the same factor to the second term to find y.

⇒ y = 16 × 3 = 48.

∴ The correct answer is 48.

Alternate Method

Given:

The ratio is 10 : 16 :: 30 : y.

Formula Used:

In a proportion a : b :: c : d, the Product of Extremes = Product of Means.

⇒ a × d = b × c

Calculations:

⇒ 10 × y = 16 × 30

⇒ 10y = 480

⇒ y = 480 ÷ 10

⇒ y = 48

∴ The correct answer is 48.

Additional Information

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a proportional increase in the other, represented as a/b = c/d.

Fourth Proportional

In the expression a : b :: c : d, 'd' is known as the fourth proportional to a, b, and c.

Mean Proportional

The mean proportional between two numbers a and b is given by √(a × b).

Statement:

E < T = D ≤ Q > Z = G 

Conclusions: 

I. T < G ⇒ False (Since Q > G and T ≤ Q but there is no direct relationship between G and T.)

II. Z > D ⇒ False (Since D ≤ Q and Q > Z but there is no direct relationship between Z and D.)

Hence, "Neither conclusion I nor II is true".

The correct answer is Jammu and Kashmir.

Key Points

• The Jammu and Kashmir Police recently started Two women's safety squads in Srinagar city.
• Each squad will comprise 5 Lady Police officers or personnel.
• Both squads will be headed by Inspector Khalida Parveen.

Additional Information

• The women's safety squad was started to help women in distress.
• Its main purpose is to ensure the security of girls, especially across schools, colleges, universities, coaching centres, and other such places.
• They would be patrolling all across the city including educational institutions, during the day.
• Heads of the Institutions have also been asked to contact the women's squad in case, any incident takes place outside their centres.

Given word: BOARD

After arranging it alphabetically it follows:

B changes to 2nd position.

O changes to 4th position.

A changes to 1st position.

R changes to 5th position.

D changes to 3rd position.

Hence, "None" of the letters will remain unchanged.

Shortcut Trick

Train from Amritsar starts 40 mins earlier (3:20 p.m. to 4:00 p.m.).

Distance covered by Amritsar train in 40 mins = 80 × (40/60) = 53.33 km.

Remaining distance at 4 p.m. = 450 − 53.33 = 396.67 km.

Time to meet = 396.67 ÷ (60 + 80) = 396.67 ÷ 140 ≈ 2.833 hours.

Distance from Delhi = Speed of Delhi train × Time = 60 × (396.67/140) = 170 km.

∴ The correct answer is 170 km.

Alternate Method

Given: Total distance = 450 km. Speed (Delhi) = 60 km/h. Speed (Amritsar) = 80 km/h. Start times = 4 p.m. and 3:20 p.m.

Formula Used: Distance = Speed × Time; Relative Speed = S₁ + S₂ (Opposite direction)

⇒ Let meeting time be t hours after 4:00 p.m.

⇒ Train from Amritsar travels for (t + 40/60) hours = (t + 2/3) hours.

⇒ Distance (Delhi) + Distance (Amritsar) = 450

⇒ 60 × t + 80 × (t + 2/3) = 450

⇒ 60t + 80t + 160/3 = 450

⇒ 140t = 450 − 53.33

⇒ 140t = 396.67

⇒ t = 396.67 ÷ 140 = 2.833 hours

⇒ Meeting distance from Delhi = 60 × 2.833 = 170 km.

∴ The correct answer is 170 km.

Additional Information

Relative Speed

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂).

Time Adjustment

Always calculate the distance covered by the early-starting object before applying the relative speed formula to the remaining distance.

Meeting Point Distance

Distance from a station = Speed of the train starting from that station × Time it traveled until the meeting point.

The logic followed here is:

$\sqrt{(First\ Number)}$ - 1 = Second Number 

Considering each analogy pair:

25 : 4 :: 49 : 6 :: 9 : 2 :: 100 :?  

$\sqrt{25}$ - 1 = 4

5 -1 = 4

4 = 4

25 : 4 :: 49 : 6 :: 9 : 2 :: 100 :?  

$\sqrt{49}$ - 1 = 6

7 - 1 = 6

6 = 6

25 : 4 :: 49 : 6 :: 9 : 2 :: 100 :? 

$\sqrt{9}$ - 1 = 2

3 - 1 = 2

2 = 2

25 : 4 :: 49 : 6 :: 9 : 2 :: 100 :? 

$\sqrt{100}$ - 1 = ?

10 - 1 = ?

9 = ?

Thus, in this analogy series question mark (?) is replaced by '9'.

Hence, the next number in the series is 9.

Given: 

Four identical bulbs are connected parallelly.

The voltage of the battery = 10 V

Current = 4 A 

Formula used:

Power = V × I

From Ohm's Law, V = I × R

Effective resistance in parallel, ${1\over R_ {eq}} = {1\over R_ {1}} +{1\over R_ {2}}+ {1\over R_ {3}} +{1\over R_ {4}}$

Calculation:

$V = I × R_{eq}\ R_{eq}= {10 \over 4}$

Resistance in each bulb(R),

 ${1\over R_ {eq}} = {1\over R_ {1}} +{1\over R_ {2}}+ {1\over R_ {3}} +{1\over R_ {4}} \ ={1\over R} +{1\over R}+ {1\over R} +{1\over R} \ ={4\over R}\ R_{eq}= {R \over 4}\ R = {10\over 4} \times 4\ R=10 :\Omega$

Thus, resistance in each bulb is 10 Ω. 

Current in each bulb, 

\(I = {V \over R} \ I={10 \over 10}\I=1:A\)

Power dissipated in the circuit, 

$P=V × I\ =10 \times 4 \=40$

Shortcut Trick The standard empirical relationship for a moderately skewed distribution is Mode = 3 Median − 2 Mean. To match the given options, subtract Median from both sides of the formula: Mode − Median = 3 Median − 2 Mean − Median Mode − Median = 2 Median − 2 Mean ⇒ Mode − Median = 2 (Median − Mean). ∴ The correct answer is Mode − Median = 2 (Median − Mean). Alternate Method Given: Statistical expressions involving Mean, Median, and Mode. Formula Used: Empirical Relationship: Mode = 3 Median − 2 Mean.

We test each expression for consistency with the empirical formula: ⇒ Let the formula be Mode = 3 Median − 2 Mean. ⇒ Rearranging the terms: Mode − Median = 3 Median − Median − 2 Mean. ⇒ Mode − Median = 2 Median − 2 Mean. ⇒ Mode − Median = 2 (Median − Mean). ⇒ This perfectly matches Expression 1. Checking other options: ⇒ Expression 3: Mode = 2 Median − Mean (Incorrect, factor should be 3 and 2). ⇒ Expression 4: 2 Mean = 2 Median − Mode ⇒ Mode = 2 Median − 2 Mean (Incorrect). ∴ The correct answer is Mode − Median = 2 (Median − Mean). Additional Information Empirical Relationship For a moderately skewed distribution, the relationship is Mode = 3 Median − 2 Mean. Symmetric Distribution In a perfectly symmetrical distribution (like the Normal Distribution), Mean = Median = Mode. Skewness Property In a positively skewed distribution, Mean > Median > Mode, while in negatively skewed, Mean < Median < Mode.

Given:

11 - {7 - 56 ÷ (2 × 3 + 1)}

Concept used:

BODMAS rule.

Calculation:

According to the BODMAS rule, we will solve the expression.

11 - {7 - 56 ÷ (2 × 3 + 1)}

= 11 - {7 - 56 ÷ (6 + 1)}

= 11 - {7 - 56 ÷ 7}

= 11 - {7 - 8}

= 11 - {- 1}

= 11 + 1

= 12

Hence, the correct answer is 12.

According to the given Venn diagram:

The total number of men is 11 

Those who are both photographers and men are 5.

Those who are both men and musician is 1. 

So, shading the portion of men by ignoring photographers and musicians as shown below,

From the above-shaded portion of the Venn diagram, we can conclude that,

Those who are NEITHER photographers NOR musicians are 6.

Hence, the correct answer is "6".

Shortcut Trick

Identify the pair where both numbers are prime or have no common factor other than 1.

51 & 119 → both divisible by 17.

65 & 91 → both divisible by 13.

59 & 97 → both are prime numbers, so their HCF is 1.

133 & 171 → both divisible by 19.

∴ The correct answer is 59, 97.

Alternate Method

Given: Pairs of numbers (51, 119), (65, 91), (59, 97), (133, 171).

Formula Used: Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

⇒ Case 1: 51 = 3 × 17 and 119 = 7 × 17. HCF = 17.

⇒ Case 2: 65 = 5 × 13 and 91 = 7 × 13. HCF = 13.

⇒ Case 3: 59 and 97 are both prime numbers. HCF = 1.

⇒ Case 4: 133 = 7 × 19 and 171 = 9 × 19. HCF = 19.

∴ The correct answer is 59, 97.

Additional Information

Definition of Co-prime Numbers

Two numbers are said to be co-prime if their only common factor is 1. They do not need to be prime themselves.

Properties of Prime Numbers

Any two distinct prime numbers are always co-prime to each other because they have no factors other than 1 and themselves.

Consecutive Numbers

Any two consecutive natural numbers (e.g., 14, 15) are always co-prime as their HCF is always 1.

The correct answer is Rajendra Prasad.

Key Points 

• The Constituent Assembly was formed for the creation of the Indian Constitution.
• The Constituent Assembly was formed in July 1946 based on the recommendations of the Cabinet Mission.
• As a prominent leader of congress, Dr. Rajendra Prasad worked with Mahatma Gandhi and other leaders.
• Dr. Rajendra Prasad was elected as the President of the Constituent Assembly.
• Dr. Rajendra Prasad worked as the longest-serving President of India (1950 to 1962).
• Rajendra Prasad was elected as the first President of India on 24th January 1950 and the vice-president was Harendra Coomar Mookerjee.
• He was one of the prominent leaders of the Indian independence movement and played a prominent role as the President of the Indian National Congress.

Additional Information 

• B. R. Ambedkar was an Indian polytheist, economist, politician, and social reformer.
• After the independence of India, he became the first Law and Justice Minister of independent India.
• He was the father of the Indian Constitution and one of the architects of the Republic of India.
• He was the chairman of the Drafting Committee.
• Lal Bahadur Shastri was born on 2nd October in 1904 at Mughalsarai, Uttar Pradesh in a simple family.
• His father name was Mr. Sharada Prasad and his mother name was Mrs. Ramdulari Devi. His father died when he was only a year old.
• He got his education from the East Central Railway Inter College in the Mughalsarai, Varanasi.
• He had joined the non-cooperation movement against British Government in 1921 by Mahatma Gandhi when he was only seventeen years old.
• He had completed his graduation from the Kashi Vidyapeeth, Varanasi in the year 1926 and was awarded the title of Shastri.
• On 11 January 1966, he died in Tashkent, USSR (now Uzbekistan) after signing the Tashkent Declaration with Pakistani President Ayub Khan.
• Jawaharlal Nehru was born on 14 November 1889 in Allahabad.
• In 1912, he participated in the Bankipur Conference as a delegate.
• In 1919, he became the Secretary of the Home Rule League of Allahabad.
• In 1916, he met Mahatma Gandhi for the first time, from whom he was greatly inspired.
• He is the first prime minister of India.

The pattern followed here is:

​

Hence, "145" will replace the question mark (?) in the given series.

The correct answer is Servants of the People Society.

Key Points

• Lala Lajpat Rai, one of the most prominent leaders of the Indian independence movement, was associated with the Servants of the People Society (Lok Sevak Mandal) since 1921.
• He founded the Servants of the People Society, a non-profit welfare organization, in Lahore in 1921.
• The society was established with the objective of serving the society in a non-political, non-sectarian manner, with its primary focus being education.
• Even after his death, the Servants of the People Society continued to function, with branches in several parts of India, carrying out charitable and social service activities.

Additional Information

The correct answer is Lead iodide.

Key Points

• When lead nitrate solution is mixed with sodium iodide solution, a yellow precipitate is formed along with potassium nitrate.
• The yellow precipitate formed is lead iodide $(PbI_2)$.
• The balanced chemical equation of this reaction is:
• \((PbNO_3)2(aq) +2NaI(aq)→PbI 2(s)+2NaNO_3(aq)\)

Additional Information

• Sodium nitrite:
• ​Sodium nitrite is an inorganic compound with the chemical formula NaNO₂.
• One of two processes - the oxidation of lower nitrogen oxides or the reduction of nitrate salts is used in the industrial manufacturing of sodium nitrite.
• Lead nitrite:
• Lead nitrite is an inorganic compound with the chemical formula Pb(NO₃)2.
• Lead nitrate is produced by reaction of lead(II) oxide with concentrated nitric acid:
• PbO + 2 HNO3(concentrated) → Pb(NO3)2↓ + H2O
• Sodium nitrate:
• Sodium nitrate is an inorganic compound with the chemical formula NaNO3.
• Some methods of preparation of sodium nitrate are:
• NaHCO3 + HNO3 → NaNO3 + H2O + CO2
• Na2CO3 + 2HNO3 → 2NaNO3 + CO2 + H2O
• NaOH + HNO3 → NaNO3 + H2O

Given:

Position of Rahul from the front end is 21^st.

Position of Ankit from the rear end is 10^th.

Number of people between Rahul and Ankit is 4.

Let us draw a diagram, based on the given information,

⇒ Total number of people in rows = Position of Rahul from front + Position of Ankit from rear end + Number of people between Rahul and Ankit

⇒ Total number of people in rows = 21 + 10 + 4

⇒ Total number of people in row = 35

Hence, "35" is the correct answer.

Shortcut Trick

Effective rate of Compound Interest for 2 years = x + y + (xy ÷ 100)

Net Interest Rate = 7 + 7 + (7 × 7 ÷ 100) = 14.49%

Compound Interest = 14.49% of ₹30,000

CI = 14.49 × 300 = ₹4,347

∴ The correct answer is ₹4,347.

Alternate Method

Given: Principal (P) = ₹30,000, Rate of Interest (R) = 7% per annum, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + R/100)ⁿ ; CI = A − P

⇒ A = 30000 × (1 + 7 ÷ 100)²

⇒ A = 30000 × (1.07)²

⇒ A = 30000 × 1.1449

⇒ A = ₹34,347

⇒ CI = 34347 − 30000

⇒ CI = ₹4,347

∴ The correct answer is ₹4,347.

Additional Information

Difference between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest is CI − SI = P × (R ÷ 100)².

Half-Yearly Compounding

If interest is compounded semi-annually, the new Rate = R ÷ 2 and the new Time period = 2 × n.

Rule of 72

To estimate the time required to double an investment with compound interest, divide 72 by the annual interest rate (T ≈ 72 ÷ R).

The logic followed here is:

(Number + 2)² = Next number 

Considering each pair and applying the logic,

1. 3 : 25 

(3 + 2)² = 25

5² = 25

25 = 25

2. 8 : 100

(8 + 2)² = 100 

10² = 100

100 = 100

3. 12 : ?

(12 + 2)² 

⇒ 14² 

⇒ 196 

Therefore, the question mark (?) will be replaced by 196.

Hence, the correct answer is "196".

Given number:  

5613743658

Given Condition:

• One is added to each odd digit.
• Two is subtracted from each even digit.

Now, the number becomes 6424824466

Now, the 2nd digit from the left is 4

Hence, the correct option is "4".

The correct answer is "Rs. 18,100 crore".

Key Points

• The Government of India has allocated 18,100 crore to the Production Linked Incentive (PLI) Scheme "National Programme on Advanced Chemistry Cell (ACC) Battery Storage" with the goal of producing fifty (50) Gigawatt hours (GWh) of ACC to boost the country's industrial capability.
• The initiative's goals include increasing value added at home and making Indian battery production more competitive on a global scale by reducing the levelized cost of production.
• The program is made to not care about the specifics of the battery technology being used, but rather the result that is being sought.

Additional Information

• Production Linked Incentive (PLI) scheme
• To boost domestic manufacturing and cut down on import bills, the union government in March 2020 introduced a PLI scheme that aims to give companies incentives on incremental sales from products manufactured in domestic units.
• The major aim of the scheme was to invite foreign investors to set up their manufacturing units in India.
• Similarly, the PLI scheme aims to promote local manufacturers to expand their manufacturing.
• Under the Scheme, companies will get incentives on incremental sales from products manufactured in domestic units.
• The scheme is implemented by the concerned ministries/departments.
• Apart from inviting foreign companies to set shop in India, the scheme also aims to encourage local companies to set up or expand existing manufacturing units.
• The Scheme has also been approved for sectors such as automobiles, pharmaceuticals, IT hardware including laptops, mobile phones & telecom equipment, white goods, chemical cells, food processing, etc.
• It will expand the present output and will further augur the achievements. For example, Textile Industry in India is one of the largest in the world and this scheme will contribute to further growth of this sector.
• PLI Scheme will provide low-cost indigenous raw materials for schemes like Smart City, Digital India mission.
• It will give a shot in the arm for the local manufacturing units of our country.
• It will create skilled manpower for the masses and reduce the unemployment ratio as these manufacturing sectors are labor-intensive.

The correct answer is '3.6 × 106'.

Key Points

• The commercial unit of energy is Kilowatt-hour (kWh).
• 1 kWh equals to 3.6 × 106 joules of energy.
• 1 kWh is the amount of energy consumed by an electrical appliance working at a steady rate of one kilowatt for one hour.
• $1\text{kWh} = 1000 :\text{Watt-hour} = 1000 :\text{W} \times (60 \times 60:\text{seconds})=1000\times 3600 :\text{J} = 3.6 \times10^6 :\text{J}$.

The correct answer is " 17^th ".

Key Points

• In 2019, Indians went to the polls to elect a new parliament, the 17th Lok Sabha.
• The Election Commission of India held elections in seven phases across the country beginning on 11 April 2019 and ending on 19 May 2019.
• One-fourth of the members of the current Lok Sabha are women.
• There are 267 first-time representatives. There are 233 people in this group who have criminal charges pending against them.
• It's estimated that 39% of members are also active in politics or social work in their professional lives.

Additional Information

• In India's general election of 2014, representatives were chosen to fill the 16th Lok Parliament. The Electoral Commission of India held the polls in a series of nine stages, beginning on April 7 and ending on May 12.
• At the general election that took place in India in 2009, members of the 15th Lok Sabha were chosen to serve the country. Dissolved by President Pranab Mukherjee on May 18, 2014

The correct answer is parallel to the axis of the solenoid.

Key Points

• Magnetic field lines in the solenoid follow a longitudinal path, i.e., parallel to its own axis.
• The longitudinal path is the result of the formation of the loop from the lines.
• The magnitude of the magnetic field inside the solenoid is uniform and is independent of length and cross-sectional area.
• The direction of the magnetic field can be known using your right hand:
• If you wrap your fingers in the direction of current, the direction in which your thumb points is the direction of the magnetic field.

The correct answer is Four.

Key Points

• Amrit Kaal is a term describing a critical time slot available to all for achieving the highest human potential.
• Amrit kaal was the term used by the union finance minister Nirmala Sitharaman in the budget speech for 2022-2023.
• Amrit kaal have four priorities:-
• PM GatiShakti, Inclusive Development
• Productivity Enhancement & Investment,
• Sunrise Opportunities
• Energy Transition, and Climate Action, and Financing of Investments.

Shortcut Trick

Check the given options directly to save time. The options represent the larger number.

If the larger number is 65 (Option 4), the smaller consecutive number must be 64.

One-fourth of the smaller number = 64 ÷ 4 = 16.

One-fifth of the larger number = 65 ÷ 5 = 13.

Difference = 16 − 13 = 3, which completely satisfies the given condition.

∴ The correct answer is 65.

Alternate Method

Given:

Two consecutive numbers are given.

One-fourth of the smaller number exceeds one-fifth of the larger number by 3.

Calculations:

Let the smaller number be x and the larger consecutive number be (x + 1).

According to the given condition:

⇒ x/4 − (x + 1)/5 = 3

Taking the LCM of the denominators (4 and 5), which is 20:

⇒ (5x − 4(x + 1)) / 20 = 3

⇒ 5x − 4x − 4 = 60

⇒ x − 4 = 60

⇒ x = 60 + 4

⇒ x = 64

The smaller number is 64.

The larger number = x + 1

⇒ 64 + 1 = 65

∴ The correct answer is 65.

Additional Information

Consecutive Numbers

Numbers that follow each other continuously in order from smallest to largest are called consecutive numbers. If the first number is n, the next consecutive numbers are n+1, n+2, etc.

Consecutive Even/Odd Numbers

For purely even or purely odd numbers, the difference between any two consecutive terms is always 2. Example: x, x+2, x+4.

Solving Linear Equations with Fractions

When solving equations involving fractions, multiplying the entire equation by the Least Common Multiple (LCM) of all denominators helps eliminate the fractions and simplifies calculation instantly.

Given word: SHIRT

Arranging the given word alphabetically:

This shows that only one position of one alphabet "T" does not change its position.​​

Hence, "One" is the correct answer.

The correct answer is Sodium.

Key Points

• Metals consist of alkali metals, alkali earth metals, transition metals, lanthanides and actinoids.
• Metals have high reactivity with water and oxygen.
• Metal is utilised to construct a variety of objects since it is strong and long-lasting.
• They are employed in the production of vehicles, satellites, kitchenware, etc.
• Metals are usually hard. Yet, some metals are not hard.
• While mercury is a liquid metal at ambient temperature, sodium and potassium are such metals that can be cut with a knife.
• Sodium is an alkali metal which belong to Group 1.

Additional Information

• Germanium:
• ​Germanium is a metalloid.
• Metalloids have intermediate properties those within metals and non-metals.
• Metalloid are used in optoelectronics, pyrotechnics, semiconductors and electronics.
• Bromine and Neon:
• ​Bromium and Neon are non-metals.
• The elements that accept or gain electrons to generate negative ions are known as non-metals.
• In their outermost shell, non-metals often have 4, 5, 6 or 7 electrons.

Alphabets and their positional values:

The logic followed here is:

The middle alphabet has been replaced with the opposite positional alphabet, and either side of the alphabet are exchanged.

NZF : FAN

And,

HSE : EHH

Similarly,

CPN : NKC

Hence, the pair of letter-cluster from among the options that follow the same logic is "CPN : NKC".

Shortcut Trick

Ratio of adjacent angles = 2 : 3

Sum of ratio parts = 2 + 3 = 5 units

Sum of adjacent angles in a parallelogram = 180°

5 units = 180° ⇒ 1 unit = 36°

Smaller angle = 2 units = 2 × 36° = 72°

∴ The correct answer is 72°.

Alternate Method

Given:

Ratio of adjacent angles = 2/3

Sum of adjacent angles = 180°

Formula Used:

In a parallelogram, the sum of any two adjacent angles is supplementary (180°).

Calculations:

Let the larger angle be x.

Then, the smaller angle = (2/3)x

⇒ x + (2/3)x = 180°

⇒ (3x + 2x) / 3 = 180°

⇒ 5x / 3 = 180°

⇒ 5x = 540°

⇒ x = 108°

⇒ Smaller angle = (2/3) × 108° = 72°

∴ The correct answer is 72°.

Additional Information

Given:

• The names of six graduate students are:
• Anudeep
• Yamuna
• Chinmayi
• Divija
• Esha
• Guna
• There are 7 days in a week.
• None of them have the exam before Chinmayi and after Yamuna

​     

• Esha has exam on Tuesday and immediately before Anudeep.

• Guna has the exam immediately after Anudeep and immediately before Divija.

     

From the above arrangement, Divija will be having an exam on Friday.

Hence, the correct answer is Divija.

The correct answer is " Nilgiri Hills ".

Key Points

• At 2,623 mts above MSL, Doddabetta is the highest Peak in Tamil Nadu and is about 10 km from Ooty bus stand.
• The name Doddabetta literally means ‘Big mountain’ in the Badugu Language, which is so in reality.
• It is at the junction of the Western and Eastern Ghats and offers beautiful vistas of the Nilgiri Hills range.
• It is surrounded by dense Sholas.
• The peaks Hecuba (2375 m), Kattadadu (2418 m) and Kulkudi (2439 m) are the three closely linked summits in the west of the Doddabetta range near to Udagamandalam

Additional Information

• Ananthagiri Hills is located in Vikarabad district, Telangana, India. The water flows from these hills to Osman Sagar, also known as Gandipet lake, and Himayathsagar.
• The Cardamom Hills or Yela Mala are a mountain range of southern India and part of the southern Western Ghats located in the Idukki district, Kerala, India.
• The Servarayan hills, with the anglicised name Shevaroy Hills, are a towering mountain range (1620 m) near the city of Salem, in Tamil Nadu state, southern India.

The correct answer is A residual amount of air remains in the lungs.

Key Points

• The process of exhaling carbon dioxide from the body is known as expiration.
• Expiration happens when the air pressure outside increases.
• During expiration, the external intercostal muscles relax and the internal intercostal muscles contract.
• The size of the thoracic cavity decreases, this relaxes the diaphragm, and the lungs get compressed.
• Also, a residual amount of air remains in the lungs which is known as residual volume.

Statements: 

All sachets are bags. 

Some bags are clocks.

No clock is a toaster. 

Venn Diagram:

<br>

Conclusion I: Some toasters are clocks. → False (As no clock is a toaster so some toasters are clocks is not possible .)

Conclusion II: No sachet is a toaster → False (As there is no direct relationship between 'sachet' and 'toaster', so we cannot conclude anything about their relation.)

Conclusion III: Some bags are Sachets. → True (This is the correct conclusion because all sachets are bags. So some bags are sachets is a definite possibility.) 

So," only Conclusion III follows."

The correct answer is Green plants capture about 10% of the solar energy that falls on leaves.

Key Points

• Plants absorb only 1-2% of the sunlight that falls on leaves, as 98-99 % of the sunlight is reflected back and converted to heat.
• Energy flow is unidirectional from the lower trophic level to the higher trophic level.
• 10% of energy is absorbed by each trophic level and the rest is lost in the environment.

Shortcut Trick

Cost of pencil = ₹1

Cost of pen = 2 × 1 = ₹2

Required cost = (3 × 2) + (2 × 1) = 6 + 2 = ₹8

∴ The correct answer is ₹8.

Alternate Method

Given:

⇒ Cost of 1 pencil = ₹1

⇒ Cost of 1 pen = 2 × (Cost of 1 pencil)

Calculations:

⇒ Cost of 1 pen = 2 × 1 = ₹2

⇒ Cost of 3 pens = 3 × 2 = ₹6

⇒ Cost of 2 pencils = 2 × 1 = ₹2

⇒ Total cost = Cost of 3 pens + Cost of 2 pencils

⇒ Total cost = 6 + 2 = ₹8

∴ The correct answer is ₹8.

Additional Information

Unitary Method

Finding the value of a single unit first and then multiplying by the required number of units to find the total value.

Ratio and Proportion

The relationship between the cost of pen and pencil can be expressed as a ratio of 2 : 1.

Linear Equations

Problems involving multiple items can be solved by forming linear equations like Ax + By = Total Cost.

Shortcut Trick

For any two squares, the ratio of sides is the square root of the ratio of their areas.

Since Perimeter is directly proportional to the Side (P = 4s), the ratio of perimeters is the same as the ratio of sides.

Ratio of Perimeters = √(Ratio of Areas) = √2 : √3

∴ The correct answer is √2 : √3.

Alternate Method

Given: Ratio of areas of two squares (A₁ : A₂) = 2 : 3.

Formula Used: Area of square = s² | Perimeter of square = 4s

⇒ Let the sides of the two squares be s₁ and s₂.

⇒ Ratio of Areas = s₁² / s₂² = 2 / 3

⇒ Taking square root on both sides: s₁ / s₂ = √2 / √3

⇒ Perimeter of first square (P₁) = 4s₁

⇒ Perimeter of second square (P₂) = 4s₂

⇒ Ratio of Perimeters = P₁ / P₂ = 4s₁ / 4s₂ = s₁ / s₂

⇒ P₁ : P₂ = √2 : √3

∴ The correct answer is √2 : √3.

Additional Information

Relationship between Dimensions

For similar figures like squares, the ratio of lengths (sides, diagonals, perimeters) is always the square root of the ratio of their areas.

Diagonal of a Square

The ratio of the diagonals of these two squares would also be the same as the ratio of their sides, which is √2 : √3.

General Scale Factor

If the side ratio is a : b, then the perimeter ratio is a : b and the area ratio is a² : b².

Shortcut Trick

Subtract the remainders from the given numbers: (50 − 1), (58 − 2), (69 − 6).

Required number = HCF of 49, 56, and 63.

HCF(49, 56, 63) = 7 (as all are multiples of 7 and no higher common factor exists).

∴ The correct answer is 7.

Alternate Method

Given: Numbers are 50, 58, 69 and remainders are 1, 2, 6.

Formula Used: Required Number = HCF[(N₁ − R₁), (N₂ − R₂), (N₃ − R₃)]

⇒ First number: 50 − 1 = 49

⇒ Second number: 58 − 2 = 56

⇒ Third number: 69 − 6 = 63

⇒ Prime factorization of 49 = 7 × 7

⇒ Prime factorization of 56 = 2 × 2 × 2 × 7

⇒ Prime factorization of 63 = 3 × 3 × 7

⇒ Common factor = 7

∴ The correct answer is 7.

Additional Information

HCF with different remainders

To find the greatest number dividing x, y, and z leaving remainders a, b, and c, calculate HCF(x−a, y−b, z−c).

HCF with same remainder

When the same remainder 'r' is left, the number is HCF(x−r, y−r, z−r). If the remainder is unknown, find HCF(|x−y|, |y−z|, |z−x|).

LCM Application

To find the smallest number divided by x, y, z leaving remainder r in each case, the number is LCM(x, y, z) + r.

The correct answer is Sulphur dioxide and sulphur trioxide.

Key Points

• The chemical formula of ferrous sulphate is FeSO₄.
• The strong heating of ferrous sulfate leads to the formation of a brown solid (ferric oxide) and 2 gases.
• This type of reaction is known as decomposition reaction.
• The balanced chemical equation of this reaction is:
• 2 FeSO₄ (s) → Fe₂O₃ (s) + SO₂ (g) + SO₃ (g)

Thus, the two gases formed are Sulphur dioxide (SO₂) and Sulphur trioxide (SO₃).

Additional Information

• Oxygen:
• Oxygen can be evolved through electrolysis of water, thermal decomposition of various oxides and oxygenic photosynthesis.
• Electrolysis: 2 H₂O → 2 H₂ + O₂
• Disulphur: 
• Disulphur is evolved when an atmosphere of COS is irradiated with UV light.
• It can also be produced when S₂Cl₂, H₂S₂, CS₂, or C₂H₄S, PSF₃ or COS are irradiated.
• Sulphur trioxide:
• Sulphur trioxide is a chemical compound with the formula SO₃.
• Sulfur dioxide in air directly oxidizes to Sulphur trioxide extremely slowly:
• SO₂ + $1 \over 2$O₂ → SO₃

The correct answer is 'K2CO3'.

Key Points

• As the pH of the solution is more than 7, it means that the salt is basic in nature.
• Out of the given salts K2CO3 is basic in nature.
• K2CO3 is an inorganic compound which forms highly alkaline solution.
• It is a salt of strong base (KOH) and weak acid (H₂CO₃), thus it is slightly basic.
• NH4Cl is an acidic salt while KCl and Na2SO4 are neutral salts.​​

​ 

Additional Information

• Na2SO4:
• Sodium sulphate is a sodium salt of sulphuric acid.
• It is white crystalline and odorless solid.
• It is the reaction product of a strong base and a strong acid.
• Thus, it has a pH of 7, i.e., it is neutral.
• KCl:
• Potassium chloride is a salt containing metal and halogen.
• KCl is also a neutral salt.
• NH4Cl: 
• ​Ammonium chloride is an inorganic compound.
• It is highly soluble in water.
• It has a pH likely between 4.6 - 6.

The correct answer is: 'Tamil' 

Key Points

• Authors of the Sangam canon, The first known written Tamil was called Sangam, variously spelled cankam, chankam, and shangam.
• It is believed that Sangam was written between the 1st and 4th centuries CE in one of three literary schools (chankams) in Madurai, India.
• According to legend, the first Tamil Sangam was presided over by the ancient Tamil leader Siddhar Agastyar and held in Madurai. The Sangam Age describes this time frame.

Additional Information

• Sangam is a term created by later academics.
• There are three Sangams mentioned in Tamil mythology:
• The Madurai
• Kapadapura
• Thenmadurai
• There was a time when the poetry of the Sangam was considered to be that of the nobility. Sangam signifies "association" when taken at face value.
• Sangam literature is the literary term for the works of ancient Tamil authors. This writing system is the oldest example of literature from South India.

The correct answer is: 'Equator'.

Key Points

• Thickness of the troposphere is greatest at the equator because heat is transported to great heights by strong convectional currents.
• The troposphere is thicker at the equator as the equator is warmer.
• The convection currents of air expand the thickness of the troposphere at the poles.
• The simple reason is the thermal expansion of the atmosphere at the equator.

Additional Information

• The rotation of the earth causes centrifugal force which is strongest near the equator and thus pushes the atmosphere to greater heights.
• The thickness of the troposphere also varies with the season.
• The troposphere is thicker in the summer and thinner in winter all around the planet.

The correct answer is: Executive Officer - Municipal Board.

Key Points

• A functionary is someone who fills a specific role in a political party or some other organization.
• The State Government shall appoint the Chairman of the Board of every municipality mentioned in the first Schedule of the Constitution of municipal board.
• This functionary elected include:
• ​Chairman- Municipal corporation
• Chairman- District Board
• Chairman- Municipal Board
• ​All of the above can be appointed by state government but not the executive officer as it is head of organisation so must not functionary elected.
• They are appointed by the state government from the Indian Administrative Service and are responsible for heading the administrative staff of the municipal corporation, implementing the decisions, and preparing its annual budget.
• The 74th Constitutional Amendment Act of 1992 strengthened existing urban local bodies as institutions of self-governance by reorganising them into three categories—Municipal Corporations/Mahanagar Palika, Municipal Councils/Nagar Palika, Nagar Panchayat/Notified Area Council.
• The Act provided for the constitution of such bodies through direct elections of representatives.

Given:

x = $0.8\bar{7}$

Calculation:

We have,

x = $0.8\bar{7}$

Above expression is rewritten as,

$x=0.87777777.....$

Multiplying both sides with 10

$10x=8.777777777......$ (1)

Multiplying both sides with 100

$100x=87.777777......$ (2)

Subtracting equation (2) from (1), we get,

$100x-10x=(87.777....)-(8.777......)$

$90x=79.00000......$

$x={79\over90}$

Hence, the value of $0.8\bar{7}$ is $79\over90$.

Given:  Rs. 3.50 : Rs. 7.50 :: Rs. 8.26 : x

Concept Used:

According to property of ratio and proportion,

a : b :: c : d

 $⇒ \large{a\over b}={c\over d}$

Calculation:

⇒ 3.5 : 7.5 :: 8.26 : x

⇒ $\large{3.5\over 7.5}={8.26\over X}$

⇒ $X = {{8.26:\times:7.5}\over 3.5}$

⇒ $X = {61.75\over 3.5}:=:17.70$

Hence, the value of x is 'Rs. 17.70'.

Given:

• Pallavi starts walking towards the north and after covering a distance she turns to the right, then again to the right.
• After covering some more distance, she again turns to the right.

Drawing the direction and distance diagram according to the given information,

Hence, now, Pallavi is facing the West direction.

Given: 

u = 20 cm

Formula used:

$P={1\over F(in:m)}$

where, P= Power of lens

F= Focal length

Calculation:

In convex lens, when image of equal size is formed the object is placed at a 2F distance (u), which means focal length of the lens is 10 cm. 

Thus, power of lens is: 

$P={1\over F(in:m)}\ ={1\over10:cm}\={1\over{10\over100}m}\=10 : \text{D}$

Hence, the power of the lens is +10D.

The correct answer is: 'Higher education'.

Key Points

• e-PGPathshala is an initiative of the MHRD under its National Mission on Education through ICT (NME-ICT) being executed by the UGC.
• It was initiated jointly by the Ministry of Human Resource Development, CIET, and NCERT, and launched in November 2015.

Additional Information

• The content and its quality is being the key component of the education system, high quality, curriculum-based, interactive e-content in 70 subjects across all disciplines of social sciences, arts, fine arts and humanities, natural & mathematical sciences, linguistics, and languages have been developed by the subject experts working in Indian universities and other R & D institutes across the country.
• Every subject has a team of principal investigators, paper coordinators, content writers,  content reviewers, Language editors, and multimedia team.

The correct answer is They have carbon-carbon single bond only.

Key Points

• A saturated carbon compound is a hydrocarbon in which all of its carbon atoms are bonded to at least 4 other atoms.
• Saturated also implies that the compound shall have no multiple carbon-carbon bonds.
• Saturated carbon compounds are those in which two carbon atoms are only linked together by a single covalent bond, typically in the shape of a ring or chain.
• The general formula of such compounds is: C_nH_{2n + 1} (where, n = number of carbon atoms in the compound).
• Thus, saturated carbon compounds have carbon-carbon single bond only.
• Example:
• ​Ethane, Propane, Butane, Octane, etc.
• Hydrocarbons with double or triple carbon-carbon bonds are known as unsaturated carbon compounds.

Given:

81 ÷ 3 - 14 ÷ 7 × 6 + 6 - 5 = ?

Decoding the information,

We have given expression as,

81 ÷ 3 - 14 ÷ 7 × 6 + 6 - 5 

After replacing the signs from left to right and using the BODMAS rule,

= 81 - 3 × 14 - 7 + 6 ÷** 6** × 5

= 81 - 3 × 14 - 7 + 1 × 5 

= 81 + 5 - 42 - 7

= 86 - 49

= 37 

Hence, “37” is the correct answer.

The correct answer is Ionic compounds conduct electricity in molten state but not in solid state.

Key Points

• Ionic compounds conduct electricity in molten state but not in solid state because in molten state their ions are free to move from place to place.
• ionic compounds cannot conduct electricity in solid state because their ions are fixed together and cannot move.

Additional Information 

• Ions of opposite charges are tightly packed together to create crystalline solids known as ionic compounds.
• Ionic compounds often result from reactions between metals and non-metals.

Shortcut Trick

Subtracting the two equations: (7 Cars + 5 Bikes) − (5 Cars + 3 Bikes) = 600 − 400

⇒ 2 Cars + 2 Bikes = 200 ⇒ 1 Car + 1 Bike = 100

From the second condition: 5 Cars + 3 Bikes = 400

⇒ 2 Cars + 3(1 Car + 1 Bike) = 400 ⇒ 2 Cars + 3(100) = 400

⇒ 2 Cars = 400 − 300 = 100 ⇒ 1 Car = Rs. 50

∴ The correct answer is Rs. 50.

Alternate Method

Given:

Cost of 7 toy cars and 5 toy bikes = Rs. 600

Cost of 5 toy cars and 3 toy bikes = Rs. 400

Formula Used:

Linear equations in two variables: ax + by = c

Let the cost of one toy car be x and one toy bike be y.

⇒ 7x + 5y = 600 --- (i)

⇒ 5x + 3y = 400 --- (ii)

Multiply equation (i) by 3 and equation (ii) by 5 to eliminate y:

⇒ 3 × (7x + 5y) = 3 × 600 ⇒ 21x + 15y = 1800 --- (iii)

⇒ 5 × (5x + 3y) = 5 × 400 ⇒ 25x + 15y = 2000 --- (iv)

Subtracting equation (iii) from equation (iv):

⇒ (25x − 21x) + (15y − 15y) = 2000 − 1800

⇒ 4x = 200

⇒ x = 200 ÷ 4

⇒ x = 50

∴ The correct answer is Rs. 50.

Additional Information

Elimination Method

A process of solving a system of equations by multiplying the equations by suitable non-zero constants so that the coefficients of one variable become equal, then adding or subtracting to eliminate that variable.

Substitution Method

Expressing one variable in terms of the other from one equation (e.g., y = (400 − 5x) ÷ 3) and substituting it into the second equation.

Linear Equation Consistency

A system of two linear equations has a unique solution if the ratio of the coefficients of x is not equal to the ratio of the coefficients of y (a₁/a₂ ≠ b₁/b₂).

The correct answer is: It is a medium of exchange.

Key Points

• Money is a medium of transaction that is recognised internationally.
• A medium of exchange and a standard of worth, money is essential.
• Prices and values are communicated through this medium.
• It serves as the primary unit of account and medium of exchange, and it moves freely between individuals and nations.

Additional Information

• In the seventh century, small metal pieces used as currency were introduced.
• About 2,700 years ago, in what is now Turkey, the first gold, silver, and bronze coins were made.
• Banknotes are the building blocks of paper currency, and the value of each note is printed right on the note itself. Almost a millennium ago, it was initially developed in China.

The correct answer is centre of curvature and pole.

Key Points

• A spherical mirror's radius of curvature is equal to the radius of the circle it is a part of.
• It can alternatively be described as the separation between the mirror's pole and its centre of curvature on the primary axis.
• The radius of curvature is factor of the mirror's curvature.
• A spherical mirror will be more curved than one with a wider radius of curvature if the former is smaller.

Additional Information

• The focal length of the mirror is defined as the distance from the pole to the primary focus.
• The radius of curvature of the lens is the separation between the primary focus and the centre of curvature

Given:

u = -(f+x)

v = -(f+y) (since concave mirror)

Formula used:

Mirror formula: ${1 \over f} = {1 \over u}+{1\over v}$

Calculation:

$\begin{aligned} & \frac{1}{f}=\frac{1}{u}+\frac{1}{v} \ & \frac{1}{-f}=\frac{1}{-(f+x)}+\frac{1}{-(f+y)} \ & \frac{1}{f}=\frac{1}{(f+x)}+\frac{1}{(f+y)} \ & \frac{1}{f}=\frac{(f+y)+(f+x)}{(f+x)(f+y)} \ & (f+x)(f+y)=f(2 f+y+x) \ & f^2+(x+y) f+x y=2 f^2+(x+y) f \ & f^2+x y=2 f^2\ & x y=f^2 \ & f=\sqrt{x y} \end{aligned}$

Therefore, the focal length f can be expressed as: $f = \sqrt{xy}$.